Dr. Steven Gold of the National Conference of State Legislatures says he expects ''many states will increase taxes in the coming months.'' John Gambill of the Federation of Tax Administrators also says he foresees ''quite a few increases next year.'' Both decline to speculate how widespread or extensive these increases might be.

Yet it seems that most governors and legislators, especially newly elected ones who ran on ''no new tax'' platforms, remain outwardly optimistic that higher levies can and will be avoided.

This is despite the fact that several states are operating either in the red or close to it - a situation due largely to slimmer-than-expected revenues, an aftereffect of the recession.

The troubled economy has lasted longer and run deeper than anticipated when fiscal 1983 budgets were first shaped, says James Mallory of the National Association of Budget Officers.

With more than six months left in most states' current fiscal periods - some of which run two years instead of one - officials in at least 25 states have attempted to shave revenue shortfalls by whittling away at their budgets.

Similar moves are under consideration, if not imminent, in at least another 10 states, says Dr. Gold. In California, as a stopgap measure, the state is borrowing $400 million from private investors to keep itself running.

A $579 million revenue shortfall in New York State may its officials to borrow operating funds for the first time since the mid-1970s. Other states where early legislative action may be needed to meet financial needs include Kentucky, Maine, Michigan, Montana, Nebraska, Oklahoma, Tennessee, and Wisconsin.

In Maine, for example, an impending $2 million deficit is exacerbated by the need to come up with an unanticipated $32 million in tax rebates mandated by a voter-approved measure on the Nov. 2 ballot. Democratic Gov. Joseph E. Brennan, who opposed the tax indexing arrangement during his successful bid for reelection, now must help pay for it.

A perhaps equally awesome squeeze confronts newly elected New Hampshire lawmakers and incoming Republican Gov. John Sununu. Unlike incumbent Democratic Gov. Hugh J. Gallen, whom he unseated, Mr. Sununu pledged to veto any measure that sets up an income or sales tax.

Besides Alaska, which two years ago repealed its income tax, New Hampshire is the only state with no retail sales or personal income taxes.

Despite $14 million in budget cuts made earlier, New Hampshire faces a projected $40 million deficit by the end of its current two-year fiscal period next June 30.

While few specific proposals for raising state taxes have surfaced in recent months, study panels and individual lawmakers in several states are weighing plans for new or increased taxes.

A special commission is holding hearings around Connecticut during the next few weeks. A proposed personal income tax is one of the revenue alternatives it's considering. The Nutmeg State is among only 10 in the nation without such a levy. Newly reelected Democratic Gov. William A. O'Neill has made it clear he wants no part of such a tax. Connecticut is facing a deficit of up to $200 million in the current fiscal year.

Although the constitutions in all states but Vermont specifically require that annual budgets be balanced, no penalties are prescribed for missing the target. Most, if not all, states made it through fiscal 1982 in the black, thanks in some instances to midyear budget-slimming and deferring some payments to fiscal 1983.

But the fiscal pinch is considerably more widespread. Some states that generally make out quite well through severance taxes on their natural resources have had problems.

Dr. Gold notes that Louisiana was among the 25 states that imposed budget reductions earlier this year to face up to lower-than-expected revenues. Other states where at least modest adjustments were made to cut spending after their budgets were approved are Alabama, Arizona, Arkansas, Colorado, Florida, Georgia , Idaho, Illinois, Indiana, Iowa, Kansas, Minnesota, Missouri, Nevada, New Hampshire, New Jersey, North Carolina, North Dakota, Ohio, Oregon, South Carolina, Utah, Virginia, and Washington.

Dealing with the money squeeze, whether through budget-slimming, higher taxes , or some other means, will be the major challenge in many states at 1983 legislative sessions. In Minnesota, New York, Ohio, and Washington, special sessions in December are being considered to deal with the problem.